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| {{DISPLAYTITLE:Propositional Equation Reasoning Systems}} | | {{DISPLAYTITLE:Propositional Equation Reasoning Systems}} |
− | * '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems InterSciWiki version].
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| '''Author: [[User:Jon Awbrey|Jon Awbrey]]''' | | '''Author: [[User:Jon Awbrey|Jon Awbrey]]''' |
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| ~ q \le r | | ~ q \le r |
| \\ | | \\ |
− | \overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)} | + | \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} |
| \\ | | \\ |
| ~ p \le r | | ~ p \le r |
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| ~ q \le r | | ~ q \le r |
| \\ | | \\ |
− | =\!=\!=\!=\!=\!=\!=\!=
| + | \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}} |
| \\ | | \\ |
| ~ p \le q \le r | | ~ p \le q \le r |
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| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \texttt{(} p \texttt{~(} q \texttt{))} | + | \texttt{(} p \texttt{ (} q \texttt{))} |
| \\[4pt] | | \\[4pt] |
− | \texttt{(} q \texttt{~(} r \texttt{))} | + | \texttt{(} q \texttt{ (} r \texttt{))} |
| \\[4pt] | | \\[4pt] |
− | \texttt{(} p \texttt{~(} r \texttt{))} | + | \texttt{(} p \texttt{ (} r \texttt{))} |
| \\[4pt] | | \\[4pt] |
− | \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))} | + | \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))} |
| \end{matrix}</math> | | \end{matrix}</math> |
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| | [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52) | | | [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52) |
| |- | | |- |
− | | <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}\!</math> | + | | <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!</math> |
| |- | | |- |
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| | [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53) | | | [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53) |
| |- | | |- |
− | | <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}\!</math> | + | | <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!</math> |
| |- | | |- |
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| | [[Image:Venn Diagram (P (R)).jpg|500px]] || (54) | | | [[Image:Venn Diagram (P (R)).jpg|500px]] || (54) |
| |- | | |- |
− | | <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}\!</math> | + | | <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!</math> |
| |- | | |- |
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| | [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55) | | | [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55) |
| |- | | |- |
− | | <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}\!</math> | + | | <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}\!</math> |
| |} | | |} |
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| ~ q \le r | | ~ q \le r |
| \\ | | \\ |
− | \overline{15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)15:22, 6 November 2016 (UTC)} | + | \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} |
| \\ | | \\ |
| ~ p \le r | | ~ p \le r |
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| ~ q \le r | | ~ q \le r |
| \\ | | \\ |
− | =\!=\!=\!=\!=\!=\!=\!=
| + | \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}} |
| \\ | | \\ |
| ~ p \le q \le r | | ~ p \le q \le r |
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| |- style="height:40px; background:#f0f0ff" | | |- style="height:40px; background:#f0f0ff" |
| | <math>p \le q \le r\!</math> | | | <math>p \le q \le r\!</math> |
− | | <math>\texttt{(} p \texttt{~(} q \texttt{))}\!</math> | + | | <math>\texttt{(} p \texttt{ (} q \texttt{))}\!</math> |
− | | <math>\texttt{(} p \texttt{~(} r \texttt{))}\!</math> | + | | <math>\texttt{(} p \texttt{ (} r \texttt{))}\!</math> |
− | | <math>\texttt{(} q \texttt{~(} r \texttt{))}\!</math> | + | | <math>\texttt{(} q \texttt{ (} r \texttt{))}\!</math> |
| |} | | |} |
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| In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that. | | In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that. |
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− | At first sight, the relationships seem easy enough to write out. Figure 75 shows how the various logical expressions are related to each other: The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication. | + | At first sight, the relationships seem easy enough to write out. Figure 75 shows how the various logical expressions are related to each other: The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication. |
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| {| align="center" border="0" cellpadding="10" | | {| align="center" border="0" cellpadding="10" |
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| ==References== | | ==References== |
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− | * [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in Parkinson, G.H.R. (ed.), ''Leibniz : Logical Papers'', Oxford University Press, London, UK, 1966. (Cf. Gerhardt, 7, p. 223). | + | * Leibniz, G.W. (1679–1686 ?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in Parkinson, G.H.R. (ed.), ''Leibniz : Logical Papers'', Oxford University Press, London, UK, 1966. (Cf. Gerhardt, 7, p. 223). |
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− | * [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]]. | + | * [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]]. |
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− | * [[Charles Peirce|Peirce, C.S.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7–8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as CP volume.paragraph. | + | * [[Charles Sanders Peirce|Peirce, C.S.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as CP volume.paragraph. |
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− | * Peirce, C.S. (1981–), ''Writings of Charles S. Peirce: A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianoplis, IN. Cited as CE volume, page. | + | * Peirce, C.S. (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN. Cited as CE volume, page. |
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− | * Peirce, C.S. (1885), "On the Algebra of Logic: A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted as CP 3.359–403 and CE 5, 162–190. | + | * Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted as CP 3.359–403 and CE 5, 162–190. |
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− | * Peirce, C.S. (c. 1886), "Qualitative Logic", MS 736. Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. | + | * Peirce, C.S. (c. 1886), “Qualitative Logic”, MS 736. Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague. |
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− | * Peirce, C.S. (1886 a), "Qualitative Logic", MS 582. Published as pp. 323–371 in ''Writings of Charles S. Peirce: A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. | + | * Peirce, C.S. (1886 a), “Qualitative Logic”, MS 582. Published as pp. 323–371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. |
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− | * Peirce, C.S. (1886 b), "The Logic of Relatives: Qualitative and Quantitative", MS 584. Published as pp. 372–378 in ''Writings of Charles S. Peirce: A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. | + | * Peirce, C.S. (1886 b), “The Logic of Relatives : Qualitative and Quantitative”, MS 584. Published as pp. 372–378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993. |
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− | * [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK. | + | * Spencer Brown, George (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK. |
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| ==See also== | | ==See also== |
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| ===Related essays and projects=== | | ===Related essays and projects=== |
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